Consider the functions $\mathbb{Z}\to\mathbb{C}$ that can be expressed as $\mathbb{C}$-linear combinations of functions of the form $g(n)=n^d\zeta^n$, where $d\geq 0$ is an integer and $\zeta$ is a root of unity. They are the functions which satisfy a homogeneous linear recurrence relation with constant coefficients such that the characteristic roots are roots of unity.
Do such functions have a name? Are they familiar things in any corner of mathematics?
A linear recurrence is often said to be degenerate if the ratio of two of its characteristic roots is a root of unity. Many arithmetic results on values of linear recurrences only hold for non-degenerate sequences, e.g., the Skolem-Mahler-Lech theorem. So although I haven't seen the term used before, it seems as if totally degenerate linear recurrence might be appropriate (although your sequences even have the individual characteristic roots as roots of unity, so maybe "extra-totally..."!).
